Topological Aspects of Spin and Statistics in Nonlinear Sigma Models

We study the purely topological restrictions on allowed spin and statistics of topological solitons in nonlinear sigma models. Taking as space the connected $d$-manifold $X$, and considering nonlinear sigma models with the connected manifold $M$ as target space, topological solitons are given by elements of $pi_d(M)$. Any topological soliton $\alpha \in \pi_d(M)$ determines a quotient $\Stat_n(X,\alpha)$ of the group of framed braids on $X$, such that choices of allowed statistics for solitons of type $\alpha$ are given by unitary representations of $\Stat_n(X,\alpha)$ when $n$ solitons are present. In particular, when $M = S^2$, as in the $O(3)$ nonlinear sigma model with Hopf term, and $\alpha \in \pi_2(S^2)$ is a generator, we compute that $\Stat_n(\R^2,\alpha) = \Z$, while $\Stat_n(S^2,\alpha) = \Z_{2n}$. It follows that phase $\exp(i\theta)$ for interchanging two solitons of type $\alpha$ on $S^2$ must satisfy the constraint $\theta = k\pi/n$, $k \in \Z$, when $n$ such solitons are present.